Properties

Label 3.18.a.b
Level $3$
Weight $18$
Character orbit 3.a
Self dual yes
Analytic conductor $5.497$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.49666262034\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{14569}) \)
Defining polynomial: \( x^{2} - x - 3642 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{14569}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 297) q^{2} + 6561 q^{3} + ( - 594 \beta + 88258) q^{4} + (3520 \beta + 191430) q^{5} + ( - 6561 \beta + 1948617) q^{6} + (29376 \beta + 12235784) q^{7} + ( - 133604 \beta + 65170116) q^{8} + 43046721 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 297) q^{2} + 6561 q^{3} + ( - 594 \beta + 88258) q^{4} + (3520 \beta + 191430) q^{5} + ( - 6561 \beta + 1948617) q^{6} + (29376 \beta + 12235784) q^{7} + ( - 133604 \beta + 65170116) q^{8} + 43046721 q^{9} + (854010 \beta - 404691210) q^{10} + ( - 128128 \beta - 493776756) q^{11} + ( - 3897234 \beta + 579060738) q^{12} + (3383424 \beta - 1259699122) q^{13} + ( - 3511112 \beta - 217782648) q^{14} + (23094720 \beta + 1255972230) q^{15} + ( - 26993736 \beta + 25305661960) q^{16} + (21313920 \beta - 17156563182) q^{17} + ( - 43046721 \beta + 12784876137) q^{18} + ( - 116754048 \beta + 40026771092) q^{19} + (196958740 \beta - 257263047540) q^{20} + (192735936 \beta + 80278978824) q^{21} + (455722740 \beta - 129851425044) q^{22} + ( - 1365533312 \beta + 148614371352) q^{23} + ( - 876575844 \beta + 427581131076) q^{24} + (1347667200 \beta + 898347630175) q^{25} + (2264576050 \beta - 817768577538) q^{26} + 282429536481 q^{27} + ( - 4675388688 \beta - 1208069610352) q^{28} + (2063000384 \beta - 235187034786) q^{29} + (5603159610 \beta - 2655179028810) q^{30} + ( - 4379766336 \beta + 1700377227296) q^{31} + ( - 15811058064 \beta + 2513249815824) q^{32} + ( - 840647808 \beta - 3239669296116) q^{33} + (23486797422 \beta - 7890201769374) q^{34} + (48693407360 \beta + 15900669077040) q^{35} + ( - 25569752274 \beta + 3799217502018) q^{36} + ( - 58058187264 \beta + 5326006090214) q^{37} + ( - 74702723348 \beta + 27196858542132) q^{38} + (22198644864 \beta - 8264885939442) q^{39} + (203822994600 \beta - 49188865789800) q^{40} + ( - 64587956096 \beta - 56688399724374) q^{41} + ( - 23036405832 \beta - 1428871953528) q^{42} + ( - 186387709824 \beta + 30818515744940) q^{43} + (281995072040 \beta - 33600387667176) q^{44} + (151524457920 \beta + 8240433801030) q^{45} + ( - 554177765016 \beta + 223188561694296) q^{46} + (61433939072 \beta - 139822820963280) q^{47} + ( - 177105901896 \beta + 166030448119560) q^{48} + (718876781568 \beta + 30234681237945) q^{49} + ( - 498090471775 \beta + 90101775230775) q^{50} + (139840629120 \beta - 112564211037102) q^{51} + (1046875513860 \beta - 374699460462052) q^{52} + ( - 343991051712 \beta - 265482019305738) q^{53} + ( - 282429536481 \beta + 83881572334857) q^{54} + ( - 1762621724160 \beta - 153660640038840) q^{55} + (279687642080 \beta + 282790173123360) q^{56} + ( - 766023308928 \beta + 262615645134612) q^{57} + (847898148834 \beta - 340353222681906) q^{58} + (2656278051328 \beta + 863762115543228) q^{59} + (1292246293140 \beta - 16\!\cdots\!40) q^{60}+ \cdots + ( - 5515490268288 \beta - 21\!\cdots\!76) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 594 q^{2} + 13122 q^{3} + 176516 q^{4} + 382860 q^{5} + 3897234 q^{6} + 24471568 q^{7} + 130340232 q^{8} + 86093442 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 594 q^{2} + 13122 q^{3} + 176516 q^{4} + 382860 q^{5} + 3897234 q^{6} + 24471568 q^{7} + 130340232 q^{8} + 86093442 q^{9} - 809382420 q^{10} - 987553512 q^{11} + 1158121476 q^{12} - 2519398244 q^{13} - 435565296 q^{14} + 2511944460 q^{15} + 50611323920 q^{16} - 34313126364 q^{17} + 25569752274 q^{18} + 80053542184 q^{19} - 514526095080 q^{20} + 160557957648 q^{21} - 259702850088 q^{22} + 297228742704 q^{23} + 855162262152 q^{24} + 1796695260350 q^{25} - 1635537155076 q^{26} + 564859072962 q^{27} - 2416139220704 q^{28} - 470374069572 q^{29} - 5310358057620 q^{30} + 3400754454592 q^{31} + 5026499631648 q^{32} - 6479338592232 q^{33} - 15780403538748 q^{34} + 31801338154080 q^{35} + 7598435004036 q^{36} + 10652012180428 q^{37} + 54393717084264 q^{38} - 16529771878884 q^{39} - 98377731579600 q^{40} - 113376799448748 q^{41} - 2857743907056 q^{42} + 61637031489880 q^{43} - 67200775334352 q^{44} + 16480867602060 q^{45} + 446377123388592 q^{46} - 279645641926560 q^{47} + 332060896239120 q^{48} + 60469362475890 q^{49} + 180203550461550 q^{50} - 225128422074204 q^{51} - 749398920924104 q^{52} - 530964038611476 q^{53} + 167763144669714 q^{54} - 307321280077680 q^{55} + 565580346246720 q^{56} + 525231290269224 q^{57} - 680706445363812 q^{58} + 17\!\cdots\!56 q^{59}+ \cdots - 42\!\cdots\!52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
60.8511
−59.8511
−65.1063 6561.00 −126833. 1.46604e6 −427163. 2.28730e7 1.67913e7 4.30467e7 −9.54488e7
1.2 659.106 6561.00 303349. −1.08318e6 4.32440e6 1.59855e6 1.13549e8 4.30467e7 −7.13934e8
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.18.a.b 2
3.b odd 2 1 9.18.a.c 2
4.b odd 2 1 48.18.a.h 2
5.b even 2 1 75.18.a.b 2
5.c odd 4 2 75.18.b.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.18.a.b 2 1.a even 1 1 trivial
9.18.a.c 2 3.b odd 2 1
48.18.a.h 2 4.b odd 2 1
75.18.a.b 2 5.b even 2 1
75.18.b.c 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 594T_{2} - 42912 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(3))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 594T - 42912 \) Copy content Toggle raw display
$3$ \( (T - 6561)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 382860 T - 1587996193500 \) Copy content Toggle raw display
$7$ \( T^{2} - 24471568 T + 36563624964160 \) Copy content Toggle raw display
$11$ \( T^{2} + 987553512 T + 24\!\cdots\!72 \) Copy content Toggle raw display
$13$ \( T^{2} + 2519398244 T + 85\!\cdots\!88 \) Copy content Toggle raw display
$17$ \( T^{2} + 34313126364 T + 23\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{2} - 80053542184 T - 18\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( T^{2} - 297228742704 T - 22\!\cdots\!20 \) Copy content Toggle raw display
$29$ \( T^{2} + 470374069572 T - 50\!\cdots\!80 \) Copy content Toggle raw display
$31$ \( T^{2} - 3400754454592 T + 37\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{2} - 10652012180428 T - 41\!\cdots\!20 \) Copy content Toggle raw display
$41$ \( T^{2} + 113376799448748 T + 26\!\cdots\!40 \) Copy content Toggle raw display
$43$ \( T^{2} - 61637031489880 T - 36\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{2} + 279645641926560 T + 19\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{2} + 530964038611476 T + 54\!\cdots\!20 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 17\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 27\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 45\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{2} - 436589918136724 T - 17\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 36\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 16\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 81\!\cdots\!04 \) Copy content Toggle raw display
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